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• #### Index 335-348

hours on end working quite happily in this "Fourier space," rarely emerging into "physical space."

But the time it takes to calculate Fourier coefficients is a problem. In fact, the development of fast computers and fast algorithms has been crucial to the pervasive, if quasi-invisible, use of Fourier analysis in our daily lives in connection with today's digital technology.

The basis for digital technology was given by Claude Shannon, a mathematician at Bell Laboratories whose Mathematical Theory of Communications was published in 1948; while he is not well known among the general public, he has been called a "hero to all communicators."7 Among his many contributions to information theory was the sampling theorem (discovered independently by Harry Nyquist and others). This theorem proved that if the range of frequencies of a signal measured in hertz (cycles per second) is n the signal can be represented with complete accuracy by measuring its amplitude 2n times a second.

This result, a direct consequence of Fourier analysis, is simple to state and not very difficult to prove, but it has had enormous implications for the transmission and processing of information. It is not necessary to reproduce an entire signal; a limited number of samples is enough.

Since the range of frequencies transmitted by a telephone line is about 4000 hertz, 8000 samples per second are sufficient to reconstitute your voice when you talk on the telephone; when music is recorded on a compact disc, about 44,000 samples a second are used. Measuring the amplitude more often, or trying to reproduce it continuously, as with old-fashioned records, does not gain anything.

Another consequence is that, in terms of octaves, more samples are needed in high frequencies than in low frequencies, since the frequency doubles each time you go up an octave: the range of frequency between the two lowest As on a piano is only 28 hertz, while the range of frequency between the two highest As is 1760 hertz. Encoding a piece of music played in the highest octave would require 3520 samples a second; in the lowest octave, 56 would be enough.

The sampling theorem opened the door to digital technology: a sampled signal can be expressed as a series of digits and transmitted as a series of on-and-off electrical pulses (creating, on the other hand, round-off errors). Your voice can even be shifted temporarily into different frequencies so that it can share the same telephone line with many other voices, contributing to enormous savings. (In 1915 a 3-minute call from coast to coast cost more than \$260 in today's dollars). In 1948 Shannon and his colleagues Bernard Oliver and John Pierce expected digital

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